# Octahedral Crystal Field Splitting Orbital Degeneracy

How are the $$\mathrm{e_g}$$ orbitals degenerate with each other?

Note: This isn't a homework question. After the semester ended (I don't go to MIT), I ended up on MIT open course-ware to watch some videos about areas of chemistry I haven't covered yet or haven't covered well. I am asking the question here because I have no other avenue in which to ask questions to other people. I'm just trying to use my time during the pandemic to build my knowledge of chemistry. I don't have a lot of understanding about this, I'm just looking for some help

It is because perfect octahedral symmetry is normally assumed; the two $$\mathrm{e_g}$$ levels and three $$\mathrm{t_{2g}}$$ are degenerate. If there was a distortion, say by lengthening both $$z$$-axis ligand positions then the $$\mathrm{e_g}$$ degeneracy would be removed as the $$\mathrm d_{z^2}$$ becomes more stable than $$\mathrm d_{x^2-y^2}$$. This happens because the $$z$$-axis ligand has more effect on $$\mathrm d_{z^2}$$ than on $$\mathrm d_{x^2-y^2}$$ orbitals simply due to its position. The $$\mathrm d_{xy}$$ orbital also increases in energy removing the degeneracy of the $$\mathrm{t_{2g}}$$ and the $$\mathrm d_{yz},$$ $$\mathrm d_{zx}$$ lowered.
The crystal field splitting is based on where the ligands (modelled as point charges) are in relation to the orbitals. In an octahedral complex, the ligands are all at 90° from each other and are placed on each of the $$x,$$ $$y,$$ $$z$$ axes. The orbitals that lie on these axes will experience the most repulsion and will rise in energy, while the orbitals between the axes $$(\mathrm{t_{2g}})$$ will lower in energy as they experience less repulsion from the ligands, and the average overall energy is maintained.
I can only assume that the degree by which the $$\mathrm{e_g}$$ orbitals are raised is the same due to the $$\mathrm d_{z^2}$$ orbital technically being a linear combination of what would have been the $$\mathrm d_{z^2-x^2}$$ and $$\mathrm d_{z^2-y^2}$$ orbitals. This means that the $$\mathrm{e_g}$$ orbitals lie on the axes to the same extent as each other, so experience the same overall repulsion from the ligands.